1. Introducción.

Este cuaderno ilustra dos técnicas de interpolación espacial: distancia ponderada inversa (IDW) y Kriging ordinario (OK). IDW es una técnica determinista. OK es probabilístico. Ambas técnicas se utilizan aquí para obtener una superficie continua de SOC a 15-30 cm a partir de muestras obtenidas de SoilGrids 250 m.

2. Configuración.

Primero, limpia la memoria:

rm(list=ls())

Luego, asegúrese de haber instalado previamente las bibliotecas necesarias.

Luego, carga las bibliotecas:

library("sp")
## Warning: package 'sp' was built under R version 4.3.2
library("terra")
## Warning: package 'terra' was built under R version 4.3.2
## terra 1.7.55
library("sf")
## Warning: package 'sf' was built under R version 4.3.2
## Linking to GEOS 3.11.2, GDAL 3.7.2, PROJ 9.3.0; sf_use_s2() is TRUE
library("stars")
## Warning: package 'stars' was built under R version 4.3.2
## Loading required package: abind
library("gstat")
## Warning: package 'gstat' was built under R version 4.3.2
library("automap")
## Warning: package 'automap' was built under R version 4.3.2
library("leaflet")
## Warning: package 'leaflet' was built under R version 4.3.2
library("leafem")
## Warning: package 'leafem' was built under R version 4.3.2
library("ggplot2")
## Warning: package 'ggplot2' was built under R version 4.3.2
library("dplyr")
## Warning: package 'dplyr' was built under R version 4.3.2
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:terra':
## 
##     intersect, union
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(curl)
## Warning: package 'curl' was built under R version 4.3.2
## Using libcurl 8.3.0 with Schannel
h <- new_handle()
handle_setopt(h, http_version = 2)

3. Leer los datos de entrada

Necesitamos leer un conjunto de datos para imitar datos del mundo real. Por lo tanto, leamos la capa SOC que descargamos de ISRIC usando la biblioteca terra:

archivo <- ("soc_igh_15_30.tif")
(soc <- rast(archivo))
## class       : SpatRaster 
## dimensions  : 1020, 967, 1  (nrow, ncol, nlyr)
## resolution  : 250, 250  (x, y)
## extent      : -8531500, -8289750, 173000, 428000  (xmin, xmax, ymin, ymax)
## coord. ref. : Interrupted_Goode_Homolosine 
## source      : soc_igh_15_30.tif 
## name        : soc_igh_15_30

Ahora, convierta los datos del SOC en porcentaje. Revise el factor de escala de SOC en el sitio web de SoilGrids (https://www.soilgrids.org/) y anótelo aquí.

!Conversion.PNG

Factor de escala de SOC
Factor de escala de SOC
soc.perc <-  soc/10

¿Cuál es el CRS de los datos del mundo real?

Parece que necesitamos una transformación de dicho CRS al conocido CRS WGS84:

geog ="+proj=longlat +datum=WGS84"
(geog.soc = project(soc.perc, geog))
## class       : SpatRaster 
## dimensions  : 1030, 998, 1  (nrow, ncol, nlyr)
## resolution  : 0.002224706, 0.002224706  (x, y)
## extent      : -76.63117, -74.41092, 1.553343, 3.844789  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : soc_igh_15_30 
## min value   :      9.578969 
## max value   :    119.267097

Convirtamos la capa SpatRaster en un objeto de estrellas:

stars.soc = st_as_stars(geog.soc)
m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      stars.soc,
      opacity = 0.8,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 8:130)
    ) 
#
m  # Print the map

4. Muestreo del mundo.

Consigamos una muestra de aprox. 500 sitios a partir de datos del mundo real utilizando una muestra ubicada aleatoriamente:

set.seed(123456)

# Random sampling of 500 points
(samples <- spatSample(geog.soc, 500, "random", as.points=TRUE))
##  class       : SpatVector 
##  geometry    : points 
##  dimensions  : 500, 1  (geometries, attributes)
##  extent      : -76.63006, -74.41203, 1.558904, 3.839228  (xmin, xmax, ymin, ymax)
##  coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
##  names       : soc_igh_15_30
##  type        :         <num>
##  values      :         38.88
##                         17.5
##                        18.18

Describa las características principales del objeto de muestra.

Ahora, necesitamos convertir el objeto spatVector en un objeto de característica simple:

(muestras <- sf::st_as_sf(samples))
nmuestras <- na.omit(muestras)

Visualicemos las muestras:

longit <- st_coordinates(muestras)[,1]
latit <- st_coordinates(muestras)[,2]
soc <- muestras$soc_igh_15_30
id <- seq(1,500,1) 
(sitios <- data.frame(id, longit, latit, soc))

Eliminemos los valores de NA:

sitios <- na.omit(sitios)
head(sitios)

Visualicemos las muestras:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      stars.soc,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 8:130)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions())
m  # Print the map

Ahora estamos listos para realizar las tareas de interpolación.

5. Interpolación

5.1 Creación del objeto gstat

Para interpolar, primero necesitamos crear un objeto de clase gstat, usando una función del mismo nombre: gstat

gstat es una biblioteca que se utiliza para realizar interpolación espacial mediante métodos de kriging y para realizar análisis geoestadísticos.

-La definición del modelo. -Los datos de calibración Según sus argumentos, la función gstat “entiende” qué tipo de modelo de interpolación queremos usar:

-Sin modelo de variograma → IDW -Modelo de variograma, sin covariables → Kriging ordinario

Vamos a utilizar tres parámetros de la función gstat:

-fórmula: la “fórmula” de predicción que especifica las variables dependientes e independientes (también conocidas como covariables) -datos: los datos de calibración (también conocidos como los datos del tren) -modelo: el modelo de variograma

5.2 Interpolación IDW (Inverse Distance Weighting)

Para interpolar SOC usando el método IDW creamos el siguiente objeto gstat, especificando solo la fórmula y los datos:

g1 = gstat(formula = soc_igh_15_30 ~ 1, data = nmuestras)

Ahora que nuestro modelo de interpolación g1 está definido, podemos usar la función de predicción para interpolar, es decir, estimar los valores de precipitación.

La función de predicción acepta:

El ráster tiene dos propósitos:

Creemos un objeto ráster con valores de celda iguales a 1:

# a simple copy
rrr = aggregate(geog.soc, 4)

La función aggregate se utiliza para cambiar la resolución del ráster al especificar un nuevo tamaño de celda. El objeto rrr ahora es un nuevo ráster con una resolución reducida

rrr
## class       : SpatRaster 
## dimensions  : 258, 250, 1  (nrow, ncol, nlyr)
## resolution  : 0.008898822, 0.008898822  (x, y)
## extent      : -76.63117, -74.40647, 1.548893, 3.844789  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : soc_igh_15_30 
## min value   :      10.94879 
## max value   :     103.19301

Prepararemos el raster para visualizacion, empezamos definiendo nuevos valores:

values(rrr) <-1

Values() se utiliza para obtener o asignar valores a un objeto ráster

Ahora para definir nuevos nombres:

names(rrr) <- "valor"

rrr sigue siendo un objeto ráster, pero ahora la variable dentro de la capa se llama “valor” en lugar de cualquier nombre original que pudiera haber tenido

rrr
## class       : SpatRaster 
## dimensions  : 258, 250, 1  (nrow, ncol, nlyr)
## resolution  : 0.008898822, 0.008898822  (x, y)
## extent      : -76.63117, -74.40647, 1.548893, 3.844789  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : valor 
## min value   :     1 
## max value   :     1
stars.rrr = st_as_stars(rrr)

Por ejemplo, la siguiente expresión interpola los valores SOC según el modelo definido en g1 y la plantilla ráster definida en stars.rrr:

## [interpolación ponderada por distancia inversa]
z1 = predict(g1, stars.rrr)
## [inverse distance weighted interpolation]

z1 se convierte en un objeto que almacena los valores interpolados resultantes de aplicar el modelo geoestadístico g1 a las ubicaciones definidas por el ráster stars.rrr

z1
## stars object with 2 dimensions and 2 attributes
## attribute(s):
##                Min.  1st Qu.   Median     Mean  3rd Qu.    Max.  NA's
## var1.pred  11.60271 32.21912 40.40281 40.86064 49.94967 90.8407     0
## var1.var         NA       NA       NA      NaN       NA      NA 64500
## dimension(s):
##   from  to offset     delta                       refsys x/y
## x    1 250 -76.63  0.008899 +proj=longlat +datum=WGS8... [x]
## y    1 258  3.845 -0.008899 +proj=longlat +datum=WGS8... [y]
sitios

Tome nota de los nombres de los dos atributos incluidos dentro del objeto z1.

Podemos crear un subconjunto solo del primer atributo y cambiarle el nombre a “soc”:

z1 = z1["var1.pred",,]
names(z1) = "soc"

Necesitamos una paleta de colores:

paleta <- colorNumeric(palette = c("orange", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")

El ráster SOC interpolado, utilizando IDW, se muestra en la siguiente figura:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      z1,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 11:55)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions()) %>%
    addLegend("bottomright", pal=paleta, values= z1$soc,
    title = "IDW SOC interpolation [%]"
    )
m  # Print the map

5.3 Interpolación correcta

Los métodos de Kriging requieren un modelo de variograma. El modelo de variograma es una forma objetiva de cuantificar el patrón de autocorrelación en los datos y asignar ponderaciones en consecuencia al realizar predicciones.

Como primer paso, podemos calcular y examinar el variograma empírico utilizando la función de variograma.

La función requiere dos argumentos:

-Fórmula: especifica la variable dependiente y las covariables, como en gstat -Datos: la capa de puntos con la variable dependiente y covariables como atributos de puntos

Por ejemplo, la siguiente expresión calcula el variograma empírico de muestras, sin covariables:

v_emp_ok = variogram(soc_igh_15_30 ~ 1, data=nmuestras)

Tracemos el variograma:

plot(v_emp_ok)

Hay varias formas de ajustar un modelo de variograma a un variograma empírico. Usaremos el más simple: ajuste automático usando la función autofitVariogram del paquete automap:

v_mod_ok = autofitVariogram(soc_igh_15_30 ~ 1, as(nmuestras, "Spatial"))

La función elige el tipo de modelo que mejor se adapta y también ajusta sus parámetros. Puede utilizar show.vgms() para mostrar los tipos de modelos de variograma.

Tenga en cuenta que la función autofitVariogram no funciona en objetos sf, por lo que convertimos el objeto en un SpatialPointsDataFrame (paquete sp).

El modelo ajustado se puede trazar con plot:

plot(v_mod_ok)

El objeto resultante es en realidad una lista con varios componentes, incluido el variograma empírico y el modelo de variograma ajustado. El componente $var_model del objeto resultante contiene el modelo real:

v_mod_ok$var_model

El modelo esférico(Sph) describe la variabilidad espacial a grandes distancias, mientras que el efecto pepita (Nug) representa la variabilidad no estructurada a distancias muy cortas. Estos dos componentes juntos forman el modelo de variograma y son esenciales en métodos de interpolación como el kriging ordinario, que utiliza la información del variograma para realizar predicciones espaciales en ubicaciones no muestreadas.

Ahora, el modelo de variograma se puede pasar a la función gstat y podemos continuar con la interpolación Kriging ordinaria:

## [usando kriging ordinario]
g2 = gstat(formula = soc_igh_15_30 ~ 1, model = v_mod_ok$var_model, data = nmuestras)
z2= predict(g2, stars.rrr)
## [using ordinary kriging]

Nuevamente, crearemos un subconjunto del atributo de valores predichos y le cambiaremos el nombre:

z2 = z2["var1.pred",,]
names(z2) = "soc"

Las predicciones de Kriging ordinario se muestran en la siguiente figura:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      z2,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 11:55)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions()) %>%
    addLegend("bottomright", pal = paleta, values= z2$soc,
    title = "OK SOC interpolation [%]"
    )
m  # Print the map

6. Evaluación de resultados

#6.1 Evaluación cualitativa Otra vista de las tres salidas de interpolación:

colores <- colorOptions(palette = c("orange", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")
m <- leaflet() %>%
  addTiles() %>%  
  addGeoRaster(stars.soc, opacity = 0.8, colorOptions = colores, group="RealWorld") %>%
  addGeoRaster(z1, colorOptions = colores, opacity = 0.8, group= "IDW")  %>%
  addGeoRaster(z2, colorOptions = colores, opacity = 0.8, group= "OK")  %>%
  # Add layers controls
  addLayersControl(
    overlayGroups = c("RealWorld", "IDW", "OK"),
    options = layersControlOptions(collapsed = FALSE)
  ) %>% 
    addLegend("bottomright", pal = paleta, values= z1$soc,
    title = "Soil organic carbon [%]"
)
m  # Print the map

6.2 Validación cruzada

Hemos estimado las superficies climáticas utilizando dos métodos diferentes: IDW y Kriging Ordinario. Aunque es útil examinar y comparar los resultados gráficamente, también necesitamos una forma objetiva de evaluar la precisión de la interpolación. De esa manera, podemos elegir objetivamente el método más preciso entre los métodos de interpolación disponibles.

En la validación cruzada Leave-One-Out nosotros:

-Tome un punto de los datos de calibración. -Haz una predicción para ese punto. -Repita para todos los puntos. Al final, lo que obtenemos es una tabla con un valor observado y un valor predicho para todos los puntos.

Podemos ejecutar la validación cruzada Leave-One-Out usando la función gstat.cv, que acepta un objeto gstat.

Al escribir el siguiente fragmento, oculte el mensaje y los resultados.

### ejecuta el siguiente código desde la consola, línea por línea
cv1 = gstat.cv(g1)
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## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
#cv2 = gstat.cv(g2)

El resultado de gstat.cv tiene los siguientes atributos:

-var1.pred: valor previsto -var1.var: varianza (solo para Kriging) -observado: valor observado -residual: observado-predicho -zscore: puntuación Z (solo para Kriging) -plegar: ID de validación cruzada

cv1 = na.omit(cv1)
cv1
## class       : SpatialPointsDataFrame 
## features    : 489 
## extent      : -76.60559, -74.4454, 1.558904, 3.839228  (xmin, xmax, ymin, ymax)
## crs         : +proj=longlat +datum=WGS84 +no_defs 
## variables   : 6
## names       :        var1.pred, var1.var,         observed,          residual, zscore, fold 
## min values  : 18.2446666493522,       NA, 11.4241218566895, -31.8039420008691,     NA,    1 
## max values  : 68.8253153187718,       NA, 90.9734115600586,  43.1597417951778,     NA,  489

Convirtamos el objeto cv1:

cv1 = st_as_sf(cv1)

Ahora, grafiquemos los residuos:

sp::bubble(as(cv1[, "residual"], "Spatial"))

Ahora, calculamos índices de precisión de predicción, como el error cuadrático medio (RMSE):

# This is the RMSE value for the IDW interpolation
sqrt(sum((cv1$var1.pred - cv1$observed)^2) / nrow(cv1))
## [1] 10.93502
cv2 = gstat.cv(g2)
## [using ordinary kriging]
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## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
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## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
cv2 = st_as_sf(cv2)

Calcule RSME para obtener resultados correctos:

# Este es el valor RMSE para la interpolación OK
sqrt(sum((cv2$var1.pred - cv2$observed)^2) / nrow(cv2))
## [1] 9.098487

La técnica de interpolación utilizada en cv2 parece ser más precisa que la técnica en cv1. Esto se basa en el valor más bajo de la Raíz del Error Cuadrático Medio (RMSE) en cv2 (9.0985) en comparación con el valor más alto en cv1 (10.93502). Un RMSE más bajo significa una mejor coincidencia entre las predicciones y los valores reales, lo que sugiere que la técnica asociada con cv2 (posiblemente Kriging Ordinario) proporciona estimaciones más precisas en el área considerada.

7. Referencia

Interpolación espacial del carbono orgánico del suelo. Disponible en https://rpubs.com/alejandrotrian/soc_interp

sessionInfo()
## R version 4.3.1 (2023-06-16 ucrt)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19045)
## 
## Matrix products: default
## 
## 
## locale:
## [1] LC_COLLATE=Spanish_Colombia.utf8  LC_CTYPE=Spanish_Colombia.utf8   
## [3] LC_MONETARY=Spanish_Colombia.utf8 LC_NUMERIC=C                     
## [5] LC_TIME=Spanish_Colombia.utf8    
## 
## time zone: America/Bogota
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] curl_5.1.0    dplyr_1.1.4   ggplot2_3.4.4 leafem_0.2.3  leaflet_2.2.1
##  [6] automap_1.1-9 gstat_2.1-1   stars_0.6-4   abind_1.4-5   sf_1.0-14    
## [11] terra_1.7-55  sp_2.1-1     
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.4       xfun_0.40          bslib_0.5.1        raster_3.6-26     
##  [5] htmlwidgets_1.6.2  lattice_0.21-8     vctrs_0.6.4        tools_4.3.1       
##  [9] crosstalk_1.2.0    generics_0.1.3     parallel_4.3.1     tibble_3.2.1      
## [13] proxy_0.4-27       spacetime_1.3-0    fansi_1.0.5        xts_0.13.1        
## [17] pkgconfig_2.0.3    KernSmooth_2.23-21 lifecycle_1.0.4    farver_2.1.1      
## [21] compiler_4.3.1     FNN_1.1.3.2        munsell_0.5.0      codetools_0.2-19  
## [25] htmltools_0.5.6    class_7.3-22       sass_0.4.7         yaml_2.3.7        
## [29] pillar_1.9.0       jquerylib_0.1.4    ellipsis_0.3.2     classInt_0.4-10   
## [33] cachem_1.0.8       tidyselect_1.2.0   digest_0.6.33      fastmap_1.1.1     
## [37] grid_4.3.1         colorspace_2.1-0   cli_3.6.1          magrittr_2.0.3    
## [41] base64enc_0.1-3    utf8_1.2.4         e1071_1.7-13       withr_2.5.2       
## [45] scales_1.2.1       rmarkdown_2.25     zoo_1.8-12         png_0.1-8         
## [49] evaluate_0.21      knitr_1.44         rlang_1.1.1        Rcpp_1.0.11       
## [53] glue_1.6.2         DBI_1.1.3          rstudioapi_0.15.0  reshape_0.8.9     
## [57] jsonlite_1.8.7     R6_2.5.1           plyr_1.8.9         intervals_0.15.4  
## [61] units_0.8-4